Problem 1
Customers arrive at Jackie's Jelly Bean Joint according to a Poisson process at the rate of 1 customer every 3 minutes where {X(t); t ≥ 0} represents the number of arrivals by time t. Jackie's opens each day at 8 AM. (Please convert all times to hours)
(a) The interarrival times are distributed as with ______mean___
(b) The probability that one customer arrives before 9:00AM is____
(c) The distribution of X(t) is____
(d) The distribution of the number of arrivals that occur between 3PM and 4:30PM is____
(e) The probability that 1 or more arrivals occur before 8:06AM is______
Note, to answer what a distribution is, you need to specify the name of the distribution and the value(s) of the parameter(s) of the distribution.
Problem 2
An athletic ticket office has two ticket agents answering incoming phone calls for ticket reser- vations. (Each agent has its own phone.) In addition, one caller can be put on "hold" until one of the agents is available to take the call. If both agent phones and the hold line are all busy, an arriving call gets a busy signal, and it is assumed that business is lost. The arriving calls occur according to a Poisson process with rate 12 calls/hour. Call lengths are exponentially distributed with average 4 minutes in length. Convert all rates to hourly rates.
(a) Find the long run probability that a caller gets an agent immediately?
(b) Find the long run probability that a caller is put on hold?
(c) Find the long run proportion of business that is lost.